8. The random variable \(X\) has a continuous uniform distribution over the interval \([ \alpha + 3,2 \alpha + 9 ]\) where \(\alpha\) is a constant.
The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)
- Show that \(\bar { X }\) is a biased estimator of \(\alpha\)
- Hence find the bias, in terms of \(\alpha\), when \(\bar { X }\) is used as an estimator of \(\alpha\)
Given that \(Y = \frac { 2 \bar { X } } { 3 } + k\) is an unbiased estimator of \(\alpha\)
- find the value of the constant \(k\)
A random sample of 8 values of \(X\) is taken and the results are as follows
4.8
5.8
6.5
7.1
8.2
9.5
9.9
10.6 - Use the sample to estimate the maximum value that \(X\) can take.
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